On a time-discrete convolution—space collocation BEM for the numerical solution of two-dimensional wave propagation problems in unbounded domains

Abstract For the numerical solution of a classical two-dimensional Dirichlet exterior problem for wave equation in time domain, we consider space-time boundary integral approach, based on its discretization by means coupling discrete convolution quadrature order 2, with continuous piecewise linear element method. The latter is either Galerkin method or collocation one, this being computationally much cheaper. efficient evaluation most integrals generated these two methods, but also subsequent analysis, first define new Gaussian quadrature. After recalling that stability and convergence have been proved method, while one only evidences properties, to (partially) fill gap, note can be obtained directly from system, after discretizing inner product using composite trapezoidal rule, which case turns out rule mentioned above $1$ node. Then, show all (positive) step-sizes $\varDelta _t$, when let $h\rightarrow 0$, independently matrix system produced properly normalized, converges, $\infty $-norm, corresponding matrix. behavior associated absolute relative errors as are $O\big (h^{\frac {5}{3}}+\varDelta _t^{-(1+\varepsilon )}h^{\frac {5+\varepsilon }{2}}[|\!\ln \varDelta _t|+|\!\ln h|]\big )$ {2}{3}}/|\!\ln _t|+\varDelta {3+\varepsilon }{2}}[1+|\!\ln h|/|\!\ln _t|]\big )$, respectively, $\varepsilon>0$ close likes $0$ $O(\cdot constant independent _t, h$. Some possible extensions investigation carried mentioned.